Optimal. Leaf size=56 \[ -\frac {2 (x+2) x^2}{3 \sqrt {x^2+x+1}}+\frac {1}{3} (2 x+5) \sqrt {x^2+x+1}-\frac {3}{2} \sinh ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {738, 779, 619, 215} \[ -\frac {2 (x+2) x^2}{3 \sqrt {x^2+x+1}}+\frac {1}{3} (2 x+5) \sqrt {x^2+x+1}-\frac {3}{2} \sinh ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right ) \]
Antiderivative was successfully verified.
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Rule 215
Rule 619
Rule 738
Rule 779
Rubi steps
\begin {align*} \int \frac {x^3}{\left (1+x+x^2\right )^{3/2}} \, dx &=-\frac {2 x^2 (2+x)}{3 \sqrt {1+x+x^2}}+\frac {2}{3} \int \frac {x (4+2 x)}{\sqrt {1+x+x^2}} \, dx\\ &=-\frac {2 x^2 (2+x)}{3 \sqrt {1+x+x^2}}+\frac {1}{3} (5+2 x) \sqrt {1+x+x^2}-\frac {3}{2} \int \frac {1}{\sqrt {1+x+x^2}} \, dx\\ &=-\frac {2 x^2 (2+x)}{3 \sqrt {1+x+x^2}}+\frac {1}{3} (5+2 x) \sqrt {1+x+x^2}-\frac {1}{2} \sqrt {3} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{3}}} \, dx,x,1+2 x\right )\\ &=-\frac {2 x^2 (2+x)}{3 \sqrt {1+x+x^2}}+\frac {1}{3} (5+2 x) \sqrt {1+x+x^2}-\frac {3}{2} \sinh ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 48, normalized size = 0.86 \[ \frac {6 x^2-9 \sqrt {x^2+x+1} \sinh ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right )+14 x+10}{6 \sqrt {x^2+x+1}} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.18, size = 47, normalized size = 0.84 \[ \frac {3 x^2+7 x+5}{3 \sqrt {x^2+x+1}}+\frac {3}{2} \log \left (2 \sqrt {x^2+x+1}-2 x-1\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 64, normalized size = 1.14 \[ \frac {19 \, x^{2} + 18 \, {\left (x^{2} + x + 1\right )} \log \left (-2 \, x + 2 \, \sqrt {x^{2} + x + 1} - 1\right ) + 4 \, {\left (3 \, x^{2} + 7 \, x + 5\right )} \sqrt {x^{2} + x + 1} + 19 \, x + 19}{12 \, {\left (x^{2} + x + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.64, size = 38, normalized size = 0.68 \[ \frac {{\left (3 \, x + 7\right )} x + 5}{3 \, \sqrt {x^{2} + x + 1}} + \frac {3}{2} \, \log \left (-2 \, x + 2 \, \sqrt {x^{2} + x + 1} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.35, size = 33, normalized size = 0.59
method | result | size |
risch | \(\frac {3 x^{2}+7 x +5}{3 \sqrt {x^{2}+x +1}}-\frac {3 \arcsinh \left (\frac {2 \left (\frac {1}{2}+x \right ) \sqrt {3}}{3}\right )}{2}\) | \(33\) |
trager | \(\frac {3 x^{2}+7 x +5}{3 \sqrt {x^{2}+x +1}}+\frac {3 \ln \left (2 \sqrt {x^{2}+x +1}-1-2 x \right )}{2}\) | \(40\) |
default | \(\frac {x^{2}}{\sqrt {x^{2}+x +1}}+\frac {3 x}{2 \sqrt {x^{2}+x +1}}+\frac {5}{4 \sqrt {x^{2}+x +1}}+\frac {\frac {5}{12}+\frac {5 x}{6}}{\sqrt {x^{2}+x +1}}-\frac {3 \arcsinh \left (\frac {2 \left (\frac {1}{2}+x \right ) \sqrt {3}}{3}\right )}{2}\) | \(61\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.11, size = 47, normalized size = 0.84 \[ \frac {x^{2}}{\sqrt {x^{2} + x + 1}} + \frac {7 \, x}{3 \, \sqrt {x^{2} + x + 1}} + \frac {5}{3 \, \sqrt {x^{2} + x + 1}} - \frac {3}{2} \, \operatorname {arsinh}\left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^3}{{\left (x^2+x+1\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\left (x^{2} + x + 1\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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